Thus far, the tie ins seem to be doing a fair job, as noted by @Ricky Spanish. I have yet to pick something up and go "Boy, I just don't know where this story fits in the grand scheme..."That’s just a production error with an obvious fix (don’t do it in a tech manual). However, neither TOS nor DSC are errors and they cannot occupy the same physical reality, so a tie-in must be clear on its parent show.
Here's the thing. If we want to push it, we can absolutely come up with ideas that encompass uniform changes. Operational authority, different commands or admiral's discretion. As noted in other tie-in material, the Enterprise is supposed to be a very modular ship, capable of being reconfigured for different missions. Perhaps it was beefed up because it was being prepped for the Klingon War, then changed back for a different mission profile. One thread I was discussing these changes in noted that the TMP full refit (this is almost an entirely new Enterprise) only took two years. If that's all it takes, then minor changes like Bridge module swap, engine swap and the like would probably take less time.
If you're willing to work at it then you can make it work. But, since it isn't real you can only bend it so much. No, this isn't a "get a life" or a "It's just a TV show" quip. It's noting that some things will not be acceptable to all fans. That's why flexibility is so important.